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Full Description
The interplay between the spectral theory of Schr|dinger
operators and probabilistic considerations forms the main
theme of these notes, written for the non-specialist reader
and intended to provide a brief and elementaryintroduction
to this field. An attempt is made to show basic ideas in
statu nascendi and to follow their evaluation from simple
beginnings through to more advanced results. The term
"genetic" in the title refers to this proceedure. The author
concentrates on 2 topics which, in the history of the
subject, have been of major conceptual importance - on the
one hand the Laplacian is a random medium and the left end
of its spectrum (leading to large deviation problems for
Brownian motion and the link to thenotion of entropy) and
on the other, Schr|dinger operators with general ergodic
potentials in one-dimensional space. Ideas and concepts are
explained in the simplest, possible setting and by means of
a few characteristic problems with heuristic arguments
preceding rigorous proofs.
Contents
Two simple examples.- The general heuristic picture.- Some known results and open problems.- Explanation of Theorem 1 and introduction to an extended Boltzmann theory of entropy.- Explanation of Theorem 2 and introduction to an extended Floquet-Weyl theory.- Conclusion.
